Ultrathin and nanowire-based GaAs solar cells

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Hung-Ling Chen

To cite this version:

Hung-Ling Chen. Ultrathin and nanowire-based GaAs solar cells. Optics / Photonic. Université Paris Saclay (COmUE), 2018. English. �NNT : 2018SACLS355�. �tel-02100507�

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Thès

e de doc

torat

NNT

:

2018SA

GaAs solar cells

Thèse de doctorat de l’Université Paris-Saclay préparée à l’Université Paris-Sud

École doctorale n◦ _{575: electrical, optical, bio-physics and engineering}

(EOBE)

Spécialité de doctorat: électronique et optoélectronique, nano- et microtechnologies

Thèse présentée et soutenue à Palaiseau, le 16 octobre 2018, par

M. Hung-Ling Chen

Composition du jury: M. Jean-Jacques Greffet

Professeur, IOGS, France Président du jury

M. Marko Topiˇc

Professeur, University of Ljubljana, Slovénie Rapporteur

M. Nicolas Chauvin

Chargé de Recherche, INL, France Rapporteur

M. Henri Mariette

Directeur de recherche, Institut Néel, France Examinateur

M. Oliver Höhn

Chercheur, Fraunhofer ISE, Allemagne Examinateur

M. Stéphane Collin

Chargé de Recherche, C2N-CNRS, France Directeur de thèse

M. Andrea Cattoni

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qw In antiquity, a God named Kua Fu determined to chase the Sun. Afterward, he almost caught the Sun while he was too thirsty to continue, draining the Yellow River and the Wei River. As he marched to search for the Great Lake, he died of dehydration at halfway. Finally, his wooden cane grew into a vast forest of peach trees.

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Acknowledgements

After several years of studying and fastidious exams on maths and physics at the classe prépara-toire and École polytechnique, I had the chance to get to know Laboraprépara-toire de Photonique et de Nanostructures (LPN, previous name of C2N)far from RER train stations, where Marcoussian people make beautiful devices and do science in a realistic way. I have been fascinated by the intriguing properties of light and the great potential of sunlight to power the worldwide electric-ity needs, so this PhD thesis ts very well my curioselectric-ity and my scientic background. Therefore, I would like to give my rst thanks to my supervisor: Stéphane Collin, who accepted me rstly as an internship student in Spring 2014, and then proposed a thesis subject for Autumn 2015. Thank you for the freedom you gave me and for the scientic supports, especially during the revision of my thesis manuscript and the preparation before the nal presentation. I apologize to have made you read pieces of the manuscript during your summer holidays when you were on the mountain! I have learned a lot from numerous interactions and discussions with you: either "constructive" light trapping for new limits with multi-resonant absorption, or "destructive" photonic bandgap narrowing of voltage penalty!

I would also like to express my special thanks to Andrea Cattonian expert on nanoimprint and nanofabrication, who helped me a lot in the clean room to improve the fabrication process. Thank you for the insightful suggestions when I struggled with the disordered slides of presenta-tion and for being the impartial "judge" between me and Stéphane! A particular thanks goes to Laurent Lombez, with whom I did the M2 internship at IRDEP in Spring 2015, just before my PhD started. I learned from the médecin des cellules solaires ("medical doctor" of solar cells in French) the practical diagnostics of solar cells (i.e. IV curves and EQE) and the useful method in luminescence characterization. I also appreciate the discussions with Jean-François Guillemoles, who always oered constructive and decisive points of view. I remember very well Erik Johnson, Martin Foldyna and Pere Roca i Cabarrocas for their clear and inspiring photovoltaic courses at École polytechnique. I would like to express my gratitude to the referees of my thesis: Prof. Marko Topi£ and Dr. Nicolas Chauvin for their great attention on reviewing the thesis, and to all the other jury members: Prof. Jean-Jacques Greet, Dr. Henri Mariette and Dr. Oliver Höhn for the kind acceptance and fruitful discussions we had. Thank you all for taking your precious time and bring up valuable questions in the PhD defense, which certainly opened interesting and new challenges for the future.

The research work in laboratories is an occasion for me to work closely with many brilliant people. First, I would like to thank Nicolas Vandamme, who taught me the methods of character-ization in our home-made VISIR dark room and guided me in the big clean room of Marcoussis. At the beginning, I knew nothing about experiments or clean room process, so it was really nice of him to teach me step by step. Alexandre Gaucher also helped me a lot to use the equipment for characterization and provided useful and practical suggestions. Later on, Julie Goard helped me to get familiar with the new VISIR setup: improved photocurrent measurement by Fourier transformation and the one sun measurement. Benoît Behaghel always gave straightforward re-marks. We collaborated closely while he was away in Japan and he even sent samples of solar cells back to France. I also appreciate the time spent working with Pierre Rale when we aligned

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I also recall a lot of people in our team, in particular PhD students who will certainly make further the scientic contributions: Romaric De Lépinau for fabricating even more ecient nanowire solar cells, Louis Gouillart for ultrathin CIGS solar cells and Thomas Bidaud for cathodoluminescence characterization. There are people I met (nearly) every Monday morning in the group meeting: Daniel Pelati, Amadeo Michaud, Ahmed Ben Slimane, Jérôme Michallon, and so on. I would like to thank Andrea Scaccabarozzi for his good humor and for oering plenty of interesting nanowire samples. The collaborations with the material department of C2N: Fabrice Oehler, Aristide Lemaître and Jean-Christophe Harmand were full of surprising results. I would like to thank Chalermchai Himwas and Maria Tchernycheva for collaborating and sharing their techniques with me in order to understand nanowire properties. I would like to acknowledge all the people whom I had the chance to work with: Alexandre Jare and Charles Renard for GaAs microcrystals; Amaury Delamarre, Maxime Giteau, Kentaroh Watanabe and Masakazu Sugiyama from NextPV Japan/France joint lab for fabrication of ultrathin GaAs solar cells; Oliver Höhn and Frank Dimroth from Frauhnofer ISE for providing high-quality III-V epitaxial layers.

I would like to give my special thanks to Christophe Dupuis for my training on the electronic microscope and for the valuable instructions on nearly every fabrication steps in the clean room! He has been very generous with his time giving me encouraging and inspiring words. Assistance provided by Nathalie Bardou on chemistry, etching and lithography was very helpful. I would like to thank all the stas who work in the clean room: Abdelmounaim Harouri for photolithog-raphy; Laurent Courant, Laetitia Leroy, Alan Durnez and Xavier Lafosse for metal and dielectric deposition. I want to thank people from the IT service, particularly Alain Péan for his kind and prompt assistance, and also people in the administration who have been taking care of my visa application, the PhD contract, the missions, and so on.

Besides my research work, I am immensely grateful to my family for unconditionally sup-porting me to study abroad and follow what I love to do. I would also like to show my warm gratitude to my high school teachers for their excellent mentorship. It has had an enormous inuence on my personal and scientic development. I recall my friends from my high school in Taiwan and the classmates I met in Taiwan or in France. They are now all around the world working in dierent elds, so that we have the chance to share very dierent experiences with each other and talk about the cultural shock we have encountered. Finally, I would like to express my deepest appreciation to my wife: Ãï (Huang Li-Wen), with whom I shared frustrating moments after failed experiments, as well as joyful moments of day-to-day life. I really valued those wonderful time. I hope we can go further and grow stronger together in life and in work.

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General introduction

Solar energy is one of the most abundant nature resources, and its conversion into electrical power source can meet the ever-increasing demand of electricity and the need for a renewable energy system. The worldwide installation of solar panels increases rapidly, and commercial photovoltaic modules of eciency over 20% are already available. For large-area use, the photovoltaic industry is looking for new solutions to further increase the eciency and decrease the cost of solar cells. One of the main trends is to use thinner and thinner Si wafers in solar photovoltaic panels. The thickness of thin-lm photovoltaics (CIGS, CdTe, III-V, etc.) is also decreased steadily to save scarce materials such as indium and tellurium. Using thinner semiconductor layers also favors the fabrication of light-weight or exible devices suited for numerous niche applications, and lead to higher radiation tolerance for space power applications. In other words, decreasing the absorber thickness in photovoltaics is a way to consume less material and to support new applications. It may also provide new routes toward high eciency photovoltaics.

Solar cells convert sunlight into electricity based on the photovoltaic eect, involving various disciplines like optics, material science and solid states physics. Recent developments take advan-tage of research advances in other optoelectronic devices, in materials, and in nanophotonics. For instance, decreasing the solar cell thickness requires nanophotonic light-trapping structures to compensate for the lower absorber volume and to maintain a high photocurrent generation. Such nanostructures can be implemented in the fabrication process of solar cells thanks to the recent technological progress of the semiconductor industry and research laboratories. The emergence of large-area and low-cost nanofabrication processes, like nanoparticle self-assembly and nano-imprint lithography, are of particular interest for the photovoltaic industry. Light connement in nanostructures results in higher photogenerated carrier densities, and may also provide new opportunities to explore novel concepts for high-eciency, like intermediate-band or hot-carrier solar cells.

Recent advances in material science are another source of innovation for the improvement of photovoltaic devices. Low-dimensional semiconductors like 2D materials, quantum structures, and nanowires, for instance, bring new solutions to create semiconductor heterostructures and to assemble various semiconductor compounds into a single photovoltaic devices. In this context, the ability to growth perfect low-dimensional crystals on lattice-mismatched substrates is of particular interest. Single semiconductor nanostructures like nanowires are also an attractive platform to investigate the fundamental mechanisms related to high carrier densities, or to the passivation and collection of carriers at the nanoscale. The understanding of these mechanisms may serve many photovoltaic technologies, for instance polycristalline thin lms made of sub-micrometer grains (with major challenges in grain boundary passivation), and hot-carrier solar cells that are made of ultrathin absorbers and selective contacts.

In the following, we focus on III-V semiconductors, and more particularly on GaAs-based solar cells and nanostructures. GaAs is chosen as a model material to explore high-eciency photovoltaic devices with a reduced volume of active material. Thanks to their high quality and the well-controlled fabrication processes, GaAs thin lms are the core of the record single-junction

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to 46%), and have been used for the most recent advances of high-eciency concepts. For these reasons, III-V are materials of choice to explore new concepts for light-trapping, for the direct growth of semiconductor nanostructures on lattice-mismatched substrates (Si in particular), or for the investigation of material properties and electronic transport in semiconductor nanostruc-tures.

In this thesis, I have studied two approaches separately. The rst one uses 200 nm-thick ultrathin GaAs absorber, corresponding to a tenfold thickness reduction as compared to state-of-the-art thin-lm GaAs solar cells. To achieve record eciency, we propose a periodic nanos-tructured back mirror for ecient light harvesting in the ultrathin GaAs layer. The second approach aims to use nanowires as the active solar cell absorber. Semiconductor nanowires con-stitute a completely dierent domain of nanotechnology. The unique shape of nanowires exhibits light-trapping properties for next-generation photovoltaics. Moreover, III-V nanowires can be grown on lattice-mismatched Si substrates instead of the more expensive GaAs or InP substrates. I mainly focus on modeling and characterization in order to understand nanowire properties and growth mechanisms. The organization of this manuscript is as follows:

In Chapter 1, I give an introduction on solar cell operation and an overview of photovoltaic research areas, and I present our motivations toward low-cost and high-eciency solar cells. Chapter2 describes essential theoretical background to understand the operation principle of a solar cell. This includes electromagnetic wave optics in one part, and physics of semiconductors in the second part. These theories provide a solid basis to design and analyze solar cell devices. Chapter3 summarizes the theory of luminescence and describes the cathodoluminescence (CL) setup at C2N. Throughout this thesis work, I use extensively the CL tool to characterize the properties of semiconductor nanowires. The concept of light emission is also governing many aspects of this thesis. For instance, the detailed balance principle of light absorption and emission leads to the fundamental limit of photovoltaic conversion eciency, which provides a guideline for researchers to improve the eciency of solar cells.

In Chapter 4, I present my work on of ultrathin GaAs solar cells. Theoretically, we show optical simulations of ultrathin solar cell structures with a metallic back grating reector and we investigate the mechanisms of multi-resonances. We also use 1D electrical transport simulations as well as a simple analytical model of resistive losses to identify the optimal design of ultrathin GaAs solar cells. Concerning experimental works, detailed fabrication steps and characterization are shown. We present the analysis of current-voltage characteristics and spectral responses, and we compare the results with theoretical calculations in order to further improve the eciency.

In the second part of the thesis work, we focus on the use of GaAs nanowires as a potential photovoltaic absorber. Toward ecient III-V nanowire solar cells, several challenges arise in the research of semiconductor nanowires. One of the issues is to characterize the doping at the nanoscale. In Chapter5, I propose an advanced contactless and quantitative doping assessment method based on the luminescence analysis using the generalized Planck's law. Hall eect mea-surements and CL meamea-surements are compared on a series of GaAs thin-lm samples at dierent doping levels in order to validate the doping assessment method by cathodoluminescence. In Chapter 6, we present CL measurements on GaAs nanowires grown in C2N. We demonstrate both p-type and n-type doping assessments in single GaAs nanowires by CL. By accident, we also discover systematically an usual wurtzite GaAs segment in single GaAs nanowires. The optical properties of undoped, Be-doped and Si-doped wurtzite GaAs are studied by CL polarimetry.

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provide a reasonable design guideline for high-eciency nanowire solar cells. At the end, we show the recent progress of nanowire solar cell performances at C2N.

I would like to detail here my contribution to the work presented in this manuscript. The opti-cal simulations of Chapter4(ultrathin solar cells) and Chapter7(nanowire array solar cells) have been performed by myself with the Reticolo code supplied by Philippe Lalanne from LP2N (Lab-oratoire Photonique, Numérique et Nanosciences) and Christophe Sauvan from IOGS (Institut d'Optique Graduate School). I adapted part of the Reticolo code to compute the rate of photo-generation in solar cells by integrating the absorbed radiation power over the solar spectrum. I also conducted electrical simulation for ultrathin solar cells using the integrated photogeneration rates as inputs to SCAPS-1D solar cell device simulator (Burgelman and coworkers, University of Gent, Belgium). For nanowire solar cells, I implemented a numerical solver with Matlab code to compute the built-in potential in nanowire core-shell junction.

In Chapter5and6, all the CL measurements and analysis presented are carried out by myself, with the help of Pierre Rale and Stéphane Collin at the beginning of my thesis work to get used to the new CL system in C2N. I developed hyperspectral data analysis and systematic CL mapping visualization with Matlab code, and I adapted the generalized Planck's law to t the whole CL spectra for quantitative doping assessment. I also proposed polarization-resolved CL experiments to investigate the optical properties of wurtzite GaAs. Epitaxial thin-lm GaAs samples were provided by Aristide Lemaître and Hall measurements on thin-lm GaAs were conducted by Ro-maric De Lépinau and Andrea Scaccabarozzi (C2N). III-V semiconductor nanowires were grown by several people in the Material Department at C2N: Fabrice Oehler, Chalermchai Himwas, Andrea Scaccabarozzi and Romaric De Lépinau. This part of the work aims to understand the fundamental material properties, and was performed in interaction with nanowire-related projects supervised by Maria Tchernycheva and Jean-Christophe Harmand.

Regarding the fabrication of ultrathin GaAs solar cells presented in Chapter 4, the epitaxy of the GaAs layers as well as double-layer MgF₂/Ta₂O₅ anti-reection coating have been done at the Fraunhofer Institute for Solar Energy Systems (ISE). I did nearly every other steps of the solar cell processes: photolithography, wet etching, metal deposition and lift-o, bonding and substrate removal. I also realized the nanoimprint lithography with the help of Andrea Cattoni. Solar cell current-voltage measurements, spectral responses (EQE) as well as data analysis were performed by myself.

For nanowire-based solar cells presented in Chapter7, I contributed mainly to the early-stage material qualities and doping characterizations. I carried out optical and electrical modeling to target the geometry and doping levels required in a core-shell junction nanowire solar cells. Con-tinuous experimental work had been done at C2N by Andrea Cattoni for patterned Si substrates using electron-beam lithography, Andrea Scaccabarozzi for epitaxial growth of III-V nanowire arrays, and Romaric De Lépinau for improved device fabrication. This work has been done within the ARN project NANOCELL for ecient GaAs nanowire solar cells and IPVF project E for III-V nanowire solar cells on Si.

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1 Introduction

Chapter content

1.1 Solar spectrum . . . 5

1.2 Solar cell operations. . . 6

1.3 Overview of photovoltaic technologies . . . 9

1.3.1 Wafer-based silicon solar cells . . . 9

1.3.2 Thin-lm and emerging technologies . . . 9

1.3.3 Motivation toward ultrathin and nanowire solar cells . . . 12

According to the International Energy Agency (IEA)1_{, in 2015, the world total primary} energy supply (TPES) is 13 647 Mtoe (million tonnes of oil equivalent, 1 toe = 11 630 kWh). The energy supply still relies on fossil fuels such as oil (31.7%), coal (28.1%) and natural gas (21.6%). Part of these primary sources are used to generate electricity. The total electricity production was 24 255 TWh in 2015, including renewable generations such as hydroelectricity production 3 978 TWh (16.4%), wind energy production 838 TWh (3.5%) and solar photovoltaic electricity production 247 TWh (1.0%). In 2017, the worldwide photovoltaic electricity production increases up to 443 TWh and the cumulative installation rises to 415 GW according to recent photovoltaics report2_{. Nowadays, transition toward renewable energy is urgent to lower the dependence on} limited fossil resources and to reduce emission of CO₂ which is responsible for greenhouse eect. Solar energy is an abundant natural resource. If we take the power density of 1000 W/m2

that the Sun illuminates the Earth, with projected cross-section of 1.275×1014_m2 _{(Earth radius}

is 6371 km), the Earth will capture the energy of 2015 world total primary energy supply in less than 2 hours! Of course, transforming eciently the sunlight into useful forms of energies such as electricity is not so easy, and we cannot cover the whole surface of the Earth by solar panels. But still, with 20% solar panel eciency to convert sunlight into electricity and 1% of the projected Earth surface, 4 days are enough to collect the total electricity production of a whole year. Solar cell researches have led to important technical progress and allowed world-wide photovoltaic electricity generations toward future sustainable energy system.

1.1 Solar spectrum

Figure1.1(a) sketches the situations of sunlight arriving at the ground surface of the Earth. We observe the solar disc with an angular diameter of about 0.53◦ _{(half-angle θ = 0.266}◦_{). For a}

solar panel oriented toward the sun, nearly all direct sunlight comes perpendicularly. Beside 1_{International Energy Agency (IEA):}_{http://www.iea.org/}

2_{Simon Philipps, Fraunhofer ISE and Werner Warmuth, PSE Conferences & Consulting GmbH (Last updated:}

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Figure 1.1 (a) Solar angles and atmospheric eects in the received sunlight on Earth, showing direct and diuse components, ground reection, minimum acceptance angle θ and solar zenith angle β (Figure adapted from Ref. [1]). (b) Reference solar spectral irradiance from American Society for Testing and Materials (ASTM G173)3_{. Red curve shows the extraterrestrial solar radiation (AM0) and black curve is} the blakebody radiation spectrum calculated at 5800 K (normalized to an integrated power of 1366 W/m2_). Blue curve is the global spectral radiation from the solar disk plus diused component (AM1.5G, integrated power 1000 W/m2_{). Green curve is the direct normal irradiance from the solar disc (AM1.5D, integrated} power 900 W/m2_{). Colored background marks the spectral positions of visible light around 380 nm to} 750 nm.

direct sunlight, diuse light can also strike on a solar panel. In sunny days, we see clear blue sky because small molecules in atmosphere scatter light at short wavelengths eciently. This is also why the sun looks yellow and becomes reddish at sunrise or sunset. When direct sunlight travels a thicker atmosphere, most blue-green part of the solar spectrum is scattered o. The cloud looks white because bigger molecules (e.g. ice/water particles and aerosols) scatter light equally in wavelengths, and may strongly attenuate sunlight in cloudy days. The light we receive at the ground will depend on time, location, weather or season and so on.

Reference sunlight radiations are dened in order to standardize performance measurement of solar cells. They are shown in Figure1.1(b). The Air Mass (AM) denes the direct light path length through the atmosphere, expressed as a ratio to the light path length at the zenith. We denote β the zenith angle of the sun, then AM = 1/cos(β). Air Mass 1.5 is chosen as standard test condition for terrestrial solar cells. AM1.5D is for direct component of sunlight (integrated power 900 W/m2_{), and AM1.5G includes for direct and diuse components (integrated power}

1000 W/m2_{). AM0 is referred to the solar radiation on top of the atmosphere for space solar cell}

evaluation. The solar radiation can also be approximated by a blackbody radiation of 5800 K, which is approximately the surface temperature of the sun (black curve in Figure1.1(b)).

1.2 Solar cell operations

Figure1.2(a) shows a typical solar cell structure. The active region that absorbs light is usually made of a crystalline semiconductor p-n junction. Here, the front or top side of the solar cell is opened to receive the sunlight, and the structure consists of n-type emitter and p-type base. Thereafter, the front contact grid is a cathode and the rear contact is an anode. The top surface is usually covered with anti-reection coating (ARC) to reduce reection of light thus maximize

3_{ASTM G173-03 spectra:} _{https://rredc.nrel.gov/solar/spectra/am1.5/ASTMG173/ASTMG173.html} 4_PVEducation: _{https://pveducation.org/}

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conduction band valence band Ec Ev Eg Efc Efv n p I (a) (b) h𝜐 h𝜐 h𝜐

Figure 1.2 (a) Typical solar cell structure using a semiconductor p-n junction (p-base and n-emitter) with front contact electrode (cathode) and rear contact electrode (anode). Under sunlight illumination, the solar cell generates electricity and power the external load (Figure adapted from PVEducation4_). (b) Schematic energy diagram of a semiconductor (bandgap Eg). A photon with energy hν < Eg is not absorbed by the semiconductor. The absorption of a photon with energy hν ≥ Eg will generate an electron-hole pair, but the excess photon energy Eg− hν is lost due to thermalization of electrons and holes to the band edges Ec and Ev, respectively. The quasi-Fermi levels Ef c and Ef v characterize the electrochemical potential of electrons and holes, respectively.

the number of photons entering the solar cell. Under illumination, the solar cell can power the external load connected to it.

Figure 1.2(b) depicts the situations when a photon arrives in the active region of a semi-conductor. The photons with energies hν smaller than the bandgap Eg of the semiconductor

cannot be absorbed. The photons with energies hν equal or higher than the bandgap Eg can

be absorbed to generate an electron-hole pair in the semiconductor. If the photon energy hν is higher than the bandgap Eg, the initially created electron (resp. hole) is above the conduction

band (resp. below the valence band), but is rapidly thermalized to the band edges. The ther-malization process is mainly due to energy lost to vibrations of the crystal lattice (phonons) and happens in a time scale of 10−12 _{s [}₂_{]. Under steady-state illumination, an excess population of}

electrons and holes is maintained in the semiconductor and two dierent quasi-Fermi levels Ff c

and Ff v are used to described the electrochemical potential of electrons and holes, respectively.

The solar cell operation is characterized by its current-voltage curve. Figure 1.3(a) gives an equivalent electrical circuit of a solar cell and Figure1.3(b) shows the current-voltage (JV ) curve (black solid line). The total electrical current I delivered by the solar cell is normalized to its surface area so we use systematically current density J with the unit of mA/cm2_{. Here we adapt}

the convention for a generator so that V > 0 and J > 0 correspond to positive power generated by the solar cell. The solar cell is modeled by several components:

• an ideal (photo-)current generator Jph,

• a rectifying diode (with diode ideality n) to ensure the photogenerated current is owing in one direction and not the opposite,

• a parallel resistance R_p taking into account the non-ideal rectifying eect or equivalently the existence of an ohmic shunt,

• a series resistance R_s for resistive losses in the solar cell.

It is assumed that the current under illumination Jlight(V ) is the sum of a constant net

photogenerated current Jph and the current Jdark(V ) measured in dark with the same bias

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V Rs Rp n Jph J (a) + − 𝐹𝐹 =𝐽𝑚𝑝𝑉𝑚𝑝 𝐽𝑠𝑐𝑉𝑜𝑐 (b) V_oc V_mp J_mp J_sc Voltage (V) Curr en t d ensi ty (mA/ cm²) Rs increases Rp decreases

Figure 1.3 (a) Equivalent electrical circuit of a solar cell, including a current generator (photocurrent Jph), a rectifying diode (ideality factor n), parallel resistance Rp and series resistance Rs. (b) Current-voltage (J-V) characteristic of a typical solar cell (black solid line). Jscis the short-circuit current density and Voc the open-circuit voltage of the solar cell. The maximum power (mp) point is where the output power JV is maximal (orange zone). Fill factor (FF) of the solar cell is the ratio of JmpVmp divided by JscVoc. The black dashed line shows the J-V characteristic of the ideal solar cell. Increasing Rs and decreasing Rp will lower the FF and result in apparently dierent slopes near Voc and Jsc, respectively.

homojunction solar cells. This is written as

J_light(V ) = Jph+ Jdark(V ) (1.1)

Note that Jdark(V ) is negative with the convention of generator, because the solar cell in dark

absorbs power from the measurement instrument. The dark current is connected to the diode characteristics n and parasitic resistances Rp and Rs through

J_dark= −J0 exp q(V + RsJdark) nkBT − 1 − V + RsJdark Rp (1.2)

where kB is the Boltzmann constant and T the absolute temperature. For an ideal solar cell,

we have an diode ideality n = 1, innite parallel resistance and zero series resistance. The current-voltage characteristic of the ideal solar cell under illumination is simply given by

J (V ) = Jph− J0 exp qV kBT − 1 . (1.3)

In Figure 1.3(b), the black dashed line is an ideal JV curve and the black solid line is a perturbed JV curve by a nite parallel resistance and non-zero series resistance. The short-circuit current density (Jsc) and the open-circuit voltage (Voc) are indicated. We denote (Jmp,Vmp) the

current and voltage of the solar cell operating at the maximum power point. The ll factor (FF) of the solar cell is dened by the maximum output power divided by the product JscVoc.

F F = JmpVmp JscVoc

. (1.4)

The eciency η of the solar cell is the power density generated at the maximum power point divided by the power density of the incident light Pin (100 mW/cm2 for 1 sun AM1.5G reference

spectrum).

η = JmpVmp Pin

= JscVocF F

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In Figure1.3(b), we can see the eect of resistances on the JV characteristics of the solar cell. Jsc and Voc are not inuenced by a small change in non-ideal resistances. A nite Rp (ohmic

shunt) gives rise to an apparent slope near Jsc, and a non-zero Rs results in a less steep slope

near Voc. Finite Rp and parasitic Rs thus aect the FF and lower the eciency of solar cells.

1.3 Overview of photovoltaic technologies

Figure 1.4 shows the up-to-date eciencies of solar cells as a function of the bandgap of the material used. The lines plot the Shockley-Queisser (SQ) limit (see Chapter3.3), which predicts the upper limit of the photovoltaic conversion eciency using a single bandgap semiconductor (solid line: 1 sun illumination, dashed line: 46 200 full concentration). The Jscof a single junction

solar cell depends on its ability to absorb photons from the solar spectrum. If absorption of one photon generates only one electron-hole pair in the semiconductor, then the maximum Jsc is

limited by the bandgap of the semiconductor.

On the other hand, the Voc of a solar cell is given by the dierence in the electrochemical

potential of electrons and holes

qVoc= Ef c− Ef c. (1.6)

With increased bandgap of the material, the upper limit of Vocis increased, but the total photon

absorption and thus Jsc decreases. As a consequence, there is a trade-o between current and

voltage that governs the fundamental limit of photovoltaic conversion eciency. With increasing sunlight intensities, a larger separation of quasi-Fermi levels leads to higher voltage, thus higher eciency.

Figure 1.5is the 2017 eciency chart of research solar cells classied by materials and types from National Renewable Energy Laboratory (NREL). The eciency progress with years is im-pressive, owning to technical eorts and understanding in material science. We summarize shortly each photovoltaic domain in the following.

1.3.1 Wafer-based silicon solar cells

Crystalline silicon (c-Si) is the most matured semiconductor technology used for the production of solar cells. Contrary to amorphous silicon (a-Si), c-Si is the crystalline forms of silicon and can be further distinguished between mono-crystalline silicon (mono-Si or single-crystal Si, continu-ous crystal) and multi-crystalline silicon (multi-Si or poly-Si, small crystals separated by grain boundaries). The state-of-art c-Si solar cell eciency is 26.7% (Kaneka) using interdigitated back contacts and high-quality a-Si:H/c-Si heterojunction passivation (Jsc = 42.65 mA/cm2,

Voc = 0.738V, F F = 0.849) [3, 4]. Commercial mono-Si solar panels with eciency between

19% to 22% are already available in the market, for example, Panasonic HIT (heterojunction with intrinsic thin-layer)6_{, LG NeON n-type mono-crystalline}7 _{and SunPower back contact modules}8_. For commercial multi-Si panels, typical eciency is found between 16% to 19%. Fabrication of multi-Si solar cells is cheaper than mono-Si solar cells because mono-Si wafers require a re-crystallization step (Czochralski process) which is very energy-consuming.

1.3.2 Thin-lm and emerging technologies

Unlike c-Si using thick wafer (160190 µm) to absorb sunlight, thin-lm technologies use direct bandgap semiconductors so that a few µm are enough to absorb sunlight. These materials include

5_{NREL, photovoltaic research:} _{https://www.nrel.gov/pv/} 6_{Panasonic solar:} _{https://panasonic.net/ecosolutions/solar/} 7_{LG solar:} _{http://www.lg-solar.com/}

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Figure 1.4 Shockley-Queisser limit of the conversion eciency for single-junction solar cells as func-tion of the bandgap. Solid line: eciency limit with AM1.5G 1 sun illuminafunc-tion. Dashed line: eciency limit with 46 200× AM1.5D full concentration. State-of-art solar cell eciencies at 1 sun are marked by points for dierent materials (Figure extracted from Ref. [1]).

Figure 1.5 Eciency evolution of best research photovoltaic cells5_{. They are classied by materials:} single- and multi-junction III-V semiconductors for high-eciency cells, crystalline silicon based solar cells, thin-lm technologies (chalcogenides, amorphous silicon) and other emerging photovoltaic cells (or-ganic solar cells, perovskite, etc.)

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copper indium gallium diselenide (CIGS), cadmium telluride (CdTe), amorphous silicon (a-Si) thin-lm III-V compounds and so on.

CIGS and CdTe

CIGS solar cells are usually fabricated using ZnO/CdS/CIGS heterojunction. Up-to-date record eciency of laboratory cell is 22.6% at 1 sun from ZSW (Germany) using heavy alkali elements post treatment for CIGS absorber [5], and later 22.9% demonstrated by Solar Frontier (Japan)9_. Because of the toxicity of cadmium element, Solar Frontier also develops CIGS solar cells with Cd-free buer layers and shows 22.0% cell eciency and near 20% eciency for mini modules at 1 sun [6]. Commercial Solar Frontier module eciency is about 16% to 19% at 1 sun. Due to scarcity of indium element, large-area CIGS solar panel may still not be competitive with c-Si modules. Indium-free chalcogenide thin-lm like CZTS (copper zinc tin sulde/selenide) is also investigated in laboratories. State-of-art CdTe thin-lm solar cells have 22.1% eciency at 1 sun demonstrated by First Solar (USA)10_{, who provides commercial modules of eciency in the 15%} to 18% ranges and may be more tolerant than c-Si under high temperature operation.

III-V semiconductors and multijunction solar cells

III-V compound semiconductors are grown by epitaxy to make high-eciency thin-lm solar cells. For example, GaAs is up-to-now the most ecient material to make a single junction solar cell, with eciency up to 28.8% at 1 sun (Jsc = 29.68 mA/cm2, Voc = 1.122V, F F =

0.865) [7]. Without being exhaustive, there are also InP solar cells with eciency of 24.2% at 1 sun (Jsc = 31.15 mA/cm2, Voc= 0.939V, F F = 0.826) [4], and GaInP with eciency of 20.8%

(Jsc = 16.0 mA/cm2, Voc= 1.455V, F F = 0.893) [8].

Because of a wide variety of III-V compounds and alloys, it is possible to grow several junctions with dierent bandgaps in a well controlled manner. Since high-energy photons tend to be absorbed near the top surface of semiconductors, stacking high-bandgap material at the top allows to absorb high-energy photons and to prevent their thermalization losses. Low-energy photons pass through the top junction and are absorbed deeper in the low-bandgap materials. These III-V multijunction solar cells provide higher eciencies beyond the SQ limit. The simplest way to contact multijunction solar cells is through a two-terminal conguration, i.e. each junction is connected in series using a tunnel junction. In this case, the operation current will be limited by the lowest current delivered by one of the junctions. The condition for current matching in all the individual junctions is critical for optimal power generation and requires careful design for the choice of material types and thicknesses. State-of-art two-junction GaInP/GaAs shows 32.8% eciency at 1 sun (LG), and three-junction InGaP/GaAs/InGaAs has 37.9% eciency at 1 sun (Sharp) [9].

Combining III-V semiconductors onto c-Si industry is a promising future technology break-through. Since III-V compounds provide better optical characteristics and also high electron mobility and high-frequency response, it is appearing not only for photovoltaics but also in other semiconductor-on-insulator (SOI) platform. Essig et al. demonstrated 1-sun eciency of III-V on Si solar cells to 32.8% (GaInP/Si) and 35.9% (GaInP/GaAs/Si), using mechanical stacking to transfer III-V cells grown on GaAs substrate onto Si bottom cells in a 4-terminal congu-ration [10]. Cariou et al. demonstrated III-V-on-silicon (GaInP/GaAs/Si) solar cells reaching 33.3% 1-sun eciency in a 2-terminal conguration by direct wafer bonding [11].

The bottleneck for III-V cells is their high cost due to slow crystal growth, expensive III-V crystalline substrates, and raw material scarcity. They are mainly used in satellites and partly in

9_{Solar Frontier:} _{http://www.solar-frontier.com} 10_{First Solar:} _{http://www.firstsolar.com/}

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terrestrial solar concentrating system. Using large-area Fresnel lens or parabolic mirror, small-area III-V multijunction cells are sucient to convert sunlight into electricity with unprecedented high eciency. Fraunhofer ISE/Soitec/CEA have shown 46% eciency of a 4-junction bonded solar cells (GaInP/GaAs;GaInAsP/GaInAs) under concentrated sunlight of 508 times the solar AM1.5D (ASTM E927-10) spectrum11_.

Perovskite solar cells

Perovskite is another new photovoltaic material with fast-rising eciencies since 2010. The crystal structure of Perovskite has a general chemical formula ABX₃ where A and B are two cations of very dierent size and X is an anion that bonds to both A and B. The most common perovskite solar cells use hybrid organic-inorganic lead or tin halide-based materials, such as methylammonium lead halides (CH₃NH₃PbX₃or MAPbX₃where X=I, Br or Cl). The advantage of perovskite solar cells is the possibility of low-temperature and atmospheric process, leads to lower fabrication costs. Perovskite solar cells with high eciency up to 22.7% at 1 sun is demonstrated by KRICT (Korea) (Jsc = 24.92 mA/cm2, Voc= 1.144V, F F = 0.796) [12].

High-bandgap perovskite is also compatible as top cell on c-Si. Monolithic perovskite/Si tandem solar cells with stabilized eciency of 23.6% is demonstrated [13]. Stability issues, light degradation and hysteresis are still problematic for hybrid perovskite materials.

Summary

Wafer-based c-Si is generally referred to as 1st generation solar cells and 2nd generation is used for thin-lm solar cells. Figure1.6shows the graph of eciency-cost of photovoltaic technologies. The dashed lines represent the constant module price per watt peak (Wp). c-Si modules have reduced the market price below 1 USD/Wp since 201312_{. This is due to improved eciency and} thinner c-Si wafers (nowadays about 160190 µm, in contrast to 300 µm in the early 2000s).

In 2017, the worldwide photovoltaic module production is estimated around 97.5 GWp. 95% of the total production is from wafer-based Si technology (62% multi-Si and 33% mono-Si) and 5% of the total production is from thin-lm technologies (CdTe, CIGS and a-Si)13_{. III-V} multijunction and concentrating photovoltaic (CPV) systems still account for very few percentage in the terrestrial photovoltaic application. Toward 3rd generation photovoltaics, which is dened as low-cost and high-eciency thin-lm solar cells, advanced concepts are investigated in research laboratories.

1.3.3 Motivation toward ultrathin and nanowire solar cells

In this thesis, we focus on ultrathin and nanowire-based GaAs solar cells. Reducing the active region of semiconductor absorber contributes to cheaper photovoltaic application by diminishing the material consumption, which can be achieved using ultrathin layers or nanowires. GaAs is the material of choice to demonstrate the possibility of high eciency. Of course, these structures can also be applied to other photovoltaic technologies. Through experimental works and theoretical investigations, we study both optical light-trapping and electronic design of GaAs solar cells for next generation high-eciency and low-cost photovoltaics.

For ultrathin GaAs solar cells, the main results are presented in Chapter4and more details of state-of-art GaAs solar cells can be found in the beginning of this chapter (Section4.1). We study 11_{Press release, Fraunhofer ISE, 1 December 2014:} _{https://www.ise.fraunhofer.de/en/press-media/}

press-releases/2014/new-world-record-for-solar-cell-efficiency-at-46-percent.html

12_{IEA technology roadmap (2014):} _{https://www.iea.org/publications/freepublications/publication/}

TechnologyRoadmapSolarPhotovoltaicEnergy_2014edition.pdf

13_{Photovoltaics report (Last updated: June 19, 2018):} _{https://www.ise.fraunhofer.de/en/publications/}

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Figure 1.6 Eciency-cost for the three generations of solar cell technology: (I) wafer, (II) thin-lms and (III) advanced concepts. The dashed lines show constant module price per watt peak (gure adapted from Ref. [14], 2006). The price of mono- and multi-crystalline silicon modules have been dropped con-siderably in recent years, and thin-lm technologies (CIGS and CdTe) continue to progress in eciency. High-eciency and low-cost 3rd generation solar cells may use ultrathin lms or nanowires for light harvesting.

advanced light management in ultrathin layers (200 nm-thick, ten-fold thinner than conventional thin-lm GaAs solar cells) and discuss its potential to achieve high eciency. Nanowires con-stitute another interesting research domain. They oer promising light concentrating property and provide the possibility of direct growth of III-V compounds on lattice-mismatched Si sub-strates, leading to potential high eciency and at the same time overcome the high cost of III-V substrates. The small sizes of nanowires make the growth and characterization of the materials dicult, hence, we develop an alternative characterization method by cathodoluminescence in Chapter5. This method is used to characterize GaAs nanowires (Chapter6) and can be applied to other materials and nanostructures. The study of GaAs nanowire solar cells is presented in Chapter 7 and more details of the state-of-art III-V nanowire solar cells can be found in the beginning of this chapter (Section7.1).

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2 Physics of Solar Cells

Chapter content

2.1 Optics . . . 15 2.1.1 Maxwell equations . . . 16 2.1.2 Reection and transmission at interface . . . 18 2.1.3 Scattering-matrix method . . . 20 2.1.4 RCWA for optical absorption . . . 22 2.2 Semiconductors. . . 23 2.2.1 Carrier concentrations . . . 23 2.2.2 Transport properties . . . 24 2.2.3 Generation and recombination . . . 26 2.2.4 Semiconductor junctions. . . 29 2.3 Summary . . . 32

This chapter is dedicated to the description of theoretical background and some notions that are useful to understand the working principle of solar cells. Wave optics is essential for our studies which focus on the light management in a reduced volume of active semiconductor absorber (Section 2.1). Interference by multiple reection in multi-layer structure is calculated using the scattering matrix method, while analysis of diraction structures in solar cells requires numerical tools. We use simulations based on the Rigorous coupled-wave analysis (RCWA) whose basic features are described.

Beside optical absorption, electronic designs of solar cells are also important for high-eciency photovoltaics. Electronic characteristics of solar cells are essentially related to the properties of semiconductors. Hence, we describe important bases of the physics of semiconductors in the second part (Section 2.2). Semiconductor p-n junction is the mostly used solar cell structure. We introduce carrier concentrations (doping), transport properties, generation and recombina-tion in semiconductors. To characterize photovoltaic materials and understand the dynamics of photogenerated carriers, important parameters such as diusion length and carrier lifetime should be considered.

2.1 Optics

Photovoltaic solar cells convert the energy of sunlight into electricity. The rst task is to absorb light as much as possible in the solar cell, therefore, we need to understand the fundamental properties of light. Ray optics is well intuitive to describe the propagation of light through space, the reection and transmission at the interface between two mediums. Light can be considered as particles moving with the velocity c0 in vacuum (c0 ≈ 2.998 × 108 m/s). Einstein interpreted

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light in quanta of discrete energy E = hν, where h is the Planck constant (h ≈ 6.626×10−34_{J· s)}

and ν is the frequency. Nowadays, these quanta are called photons.

On the other hand, light also shows wave character when we look at the experience of double slit interference or the diraction eect occurred when light encounters an obstacle or a small aperture. Solar cells are usually made of thin-lm semiconductors. Hence, the wave nature of light emerges, especially when we deal with light-trapping in small volume of semiconductor in presence of nanostructures. The classical theory of electromagnetism is used to describe optical phenomena in these cases. We will adapt both wave-particle descriptions depending on the practical situation. To summarize, the photon energy E and momentum p are related by E = pc0. The energy and momentum of a photon depend only on its frequency ν or inversely,

on its wavelength λ:

E = ~ω = hc0 λ p = ~k

(2.1) where ω = 2πν is the angular frequency and ~ = h/2π is the reduced Planck constant. k is the wavevector, where the wavenumber is k = |k| = 2π/λ, and the direction of k indicates the propagation direction of light. In addition, the photon also carries a quantity called spin angular momentum of ±~. In the following, we give the essential notions of electromagnetism relevant in studying light management in solar cells.

2.1.1 Maxwell equations

We consider electromagnetic radiation in matter, in the framework of linear optics. Some rea-soning and remarks follow the reference book Semiconductor Optics [15]. The four Maxwell equations couple the electric and magnetic elds to their sources, i.e. electric charges and cur-rent densities. They are given by

∇ · D(r, t) = ρ(r, t) ∇ × E(r, t) = −∂B(r, t) ∂t ∇ · B(r, t) = 0 ∇ × H(r, t) = ∂D(r, t) ∂t + j(r, t) (2.2)

where r and t denote location and time, respectively. D is the electric displacement, E the electric eld, B the magnetic induction and H the magnetic eld. ρ is the free charge density and j is the free current density.

The electric displacement is related to the electric eld via the constitutive relation

D = εrε0E (2.3)

where εr is the relative permittivity of the medium in which the electric elds are observed

and ε0 ≈ 8.854 × 10−12 F/m is the vacuum permittivity. The relative permittivity arises from

the polarizability of the medium that expresses the linear dependence of the density of induced electric dipole moments with the electric eld. In vacuum, we have εr = 1. In general εr > 1

for semiconductors and dielectric materials, and is frequency-dependent. A scalar value of εr is

sucient to describe an isotropic medium, in which the induced dipole moment is independent of the direction. In general, εr is a tensor in an anisotropic material (e.g. hexagonal wurtzite

structure of several semiconductors). This is called birefringence or dichroism because the optical properties depend on the polarization and propagation direction of light. Similarly, the magnetic induction is related to the magnetic eld via the constitutive relation

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where µr is the relative permeability of the medium and µ0= 4π × 10−7H/mis the permeability

of vacuum. We may assume all the important materials for solar cells to be non-magnetic, i.e. µr ≡ 1, in the usual range of the electromagnetic spectrum, so that the magnetization of the

medium has negligible inuence on the optical properties. Electromagnetic wave equation

To derive the wave equation that governs the propagation of light, we consider a homogeneous and isotropic medium and assume that there is no macroscopic free space charges: ρ(r, t) ≡ 0 in Maxwell equations. The current j is driven by the electric eld

j = σE (2.5)

where σ is the conductivity of the material. For intrinsic and weakly doped semiconductors, the carrier density is small and consequently σ is small as well. Then the inequality holds: |j| = |σE| |ωD|. In the following, we consider only this case and neglect j in Maxwell equations. For heavily doped semiconductors this is no longer valid and σ will have some inuence on the optical properties at least in the infrared (IR) region of the electromagnetic spectrum [15]. The wave equation for electric eld (and similar for magnetic eld) is written

∆E = εr c2 0

∂2_E

∂t2 (2.6)

where ∆E ≡ ∇2_{E ≡ ∇(∇ · E) − ∇ × (∇ × E)} _{is the vector Laplacian of a vector eld E.}

c0 = 1/(µ0ε0) is the propagation speed of the wave in free space (vacuum).

Properties of plane harmonic waves

The simplest solution to the wave equation is the plane harmonic wave, where light of constant angular frequency ω propagates in the direction of k. We express the eld using complex values, but keeping in mind that the physical magnitude is the real part of the complex eld.

E(r, t) = E0exp (i(k· r − ωt)) (2.7)

The time-harmonic dependence of the elds is taken as exp(−iωt) and complex values are used throughout the whole mathematical manipulations. In this way, the time derivative becomes multiplication by −iω on the complex amplitude of elds, and the nabla operator ∇ is transformed by applying ik. We can derive that the electromagnetic wave is transverse: E and H are perpendicular to k and are perpendicular to each other. Moreover, the plane wave should fulll the following dispersion relation:

k2 = ω

2

c2₀εr(ω). (2.8)

In general, the relative permittivity εr(ω), also called dielectric function, is complex. For the

square root of εr we introduce a new quantity ˜n(ω), called the complex index of refraction.

˜

n(ω) = n(ω) + iκ(ω) =pεr(ω) (2.9)

The real part of ˜n is usually called refractive index in connection with Snell's law of refraction and the imaginary part is called extinction coecient. Here the wavevector is also complex

k = (n + iκ)k0= k0+ ik00 (2.10)

with k0= 2π/λ0and λ0is the wavelength in vacuum. In this way exp(ik· r) = exp(ik0· r)exp(−k00· r).

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x z θ_t r θ i θ kr k_i k_t θt E_i Hi θr E_r H_r E_t Ht n n1₂ x z θ_t r θ i θ k_r ki k_t θt E_i H_i r θ E_r Hr Et H_t n n1₂

(a) TE (s-polarization) (b) TM (p-polarization) i

θ θ_i

Figure 2.1 Scheme of light, reected and transmitted (refracted) at an interface between two isotropic media for two orthogonal, linear polarization: (a) transverse electric (TE or s-polarization) and (b) transverse magnetic (TM or p-polarization).

describes the spatial oscillation of light in matter. The amplitude of elds decreases in direction of k00 _{so κ(ω) describes the damping of the wave in the direction of propagation (for example}

z). The light intensity I(z) is proportional to the square of the eld amplitude. In connection with the Beer-Lambert law I(z) = I(0)exp(−αz), the absorption coecient α is related to the imaginary part of the complex index of refraction

α(ω) = 2ω c0

κ(ω) = 4π λ0

κ(ω). (2.11)

A quantity that is often used to judge the absorptivity of a material at a certain wavelength is the penetration depth δ = 1/α. At this depth, the intensity has decayed to a fraction 1/e of the initial value. For example, the penetration depth of GaAs is about 100 nm at λ0= 500nm, and

increases rapidly for longer wavelength of light. We arrive at a penetration depth of 740 nm at λ0 = 800nm.

2.1.2 Reection and transmission at interface

In this section, we consider the reection and transmission (refraction) of light at the plane interface between two media. We assume homogeneous and isotropic media described by the complex index of refraction ˜n1 and ˜n2. We rst recall the boundary conditions imposed by the

Maxwell equations between two media, then we derive the Fresnel equations for reection and transmission of light.

Boundary conditions

Maxwell equations are applied in each medium 1 and 2. At the interface, boundary conditions are obtained by two general laws of vector analysis, that is the law of Gauss and law of Stokes. We note n12 normal unit vector from medium 1 to medium 2. These conditions are

(D2− D1) · n12= σs

n12× (E2− E1) = 0

(B2− B1) · n12= 0

n12× (H2− H1) = js

(2.12)

where σs is the surface charge and js is the surface current density between the media. We will

drop these two terms for the general solar cell application. Therefore, the normal component of the eld D and B is continuous across the interface, and so is the tangential component of the eld E and H.

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Law of reection and refraction

As illustrated in Figure 2.1, the plane of incidence is the plane dened by the interface normal and the wavevector of the incident wave. A part of the incident light is reected, with the angle of the reected light θr is equal to the incident angle θi. The other part of light enters medium

2, where the angle of the refracted light θtis related to θi via Snell's law of refraction

n1sin(θi) = n2sin(θt) (2.13)

An important consequence of Snell's law is total internal reection. If n1 > n2, there is a critical

angle θc at which light can no longer be transmitted to medium 2:

sin(θc) =

n2

n1

. (2.14)

For θi > θc, there is a totally reected beam but no longer a transmitted one. The reected wave

has the same intensity as the incident one. However, the boundary conditions require nite eld amplitudes in medium 2. An evanescent wave exists which propagates parallel to the surface and its amplitudes decay exponentially in the direction normal to the interface over a distance of a few wavelengths [15]. For a GaAs-air interface (nGaAs ≈ 3.68 at λ0 = 800nm), we nd

θc≈ 15.8◦.

Fresnel equations

For weak absorption media (i.e. |κ| |n|), relations between the magnitudes of the incident, reected and transmitted elds are given by Fresnel equations. These equations can be obtained by expressing the eld in each medium with plane waves, then using the boundary conditions. For strong absorption material like metal or semiconductors in certain frequency ranges, Fresnel equations are still valid but the angles and index of refraction are complex values and do not have obvious geometrical interpretation.

We have to distinguish between perpendicular and parallel polarized light. In the rst case, the electric eld is perpendicular to the plane of incidence and is called transverse electric (TE) polarization or s-polarization (senkrecht is German for perpendicular). The Fresnel equations are r_{12, TE}≡ Er Ei TE = n1cos(θi) − n2cos(θt) n1cos(θi) + n2cos(θt) t_{12, TE}≡ Et Ei TE = 1 + r_{12, TE} (2.15) In the case of electric eld parallel to the plane of incidence, we call this transverse magnetic (TM) polarization or p-polarization. The Fresnel equations are dierent from the case for TE polarization r_{12, TM}≡ Er Ei TM = n1cos(θt) − n2cos(θi) n1cos(θt) + n2cos(θi) t_{12, TM}≡ Et Ei TM = n1 n2 (1 − r_{12, TM}) (2.16) The measurable quantities are the reectivity R and the transmittance T , which are given by the ratio of the square of the electric eld. In the weak absorption regime, we can see that |r_{12, TM}|can go to zero at a certain incident angle θB, known as Brewster angle. The condition

for |r12, TM| = 0 is n1cos(θt) = n2cos(θi). Apart from n1 = n2, we have θi+ θt = 90◦, i.e. at

the incident angle where the reected and refracted beams would propagate perpendicularly to each other. For θi = θB, only the component polarized perpendicularly to the plane of incidence

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substrate x z 𝜃0 incident light … 𝐸𝑗− 𝐸′_𝑗− 𝐸′𝑗−1+ 𝐸_𝑗+1+ 𝑛_𝑗 𝑛_𝑗+1 𝑛_𝑗−1 𝑛0 𝑛_𝑁 ℎ_𝑗 ℎ_𝑗+1 ℎ_𝑗−1 𝜃𝑗 𝜃𝑗−1 𝜃𝑗 𝜃𝑗+1 … … 𝐸′𝑗−1− 𝐸_𝑗+ 𝐸′_𝑗+ 𝐸_𝑗+1− 𝐸𝑁− 𝐸′₀− reflection 𝐸′₀+

Figure 2.2 General multi-layer system composed of (N-1) layers with N parallel interfaces between a semi-innite medium (optical index n0) and a semi-innite substrate (optical index nN). Each layer j (j = 1, 2, .., N − 1) has a thickness hj and is characterized by its complex optical index nj. The electric eld at the vicinity of the interface is denoted by the subscript j for the layer j and by the superscript + (resp. −) for the upward (resp. downward) propagating eld. A prime is used for the eld at the bottom of the layer (above an interface).

If we send light on the interface polarized dierently than TE or TM to the plane of incidence, we can always decompose it into two components with the above orientations, then calculate their reected or transmitted amplitudes and superpose them again. For unpolarized light, we can simply take the mean values of the two polarizations. For the reectivity that is R = (|r_{12, TE}|2 + |r_{12, TM}|2)/2. At normal incidence θi = 0 we have R independently for the two

polarizations: R = n1− n2 n1+ n2 2 . (2.17)

Due to high refractive index of typical semiconductors (34 in general), the reectivity in at air-semiconductor interface is high. For example, the reectivity at air-GaAs interface is 33% at λ0 = 800nm. This results in approximately one third of the solar energy not being absorbed

by the solar cell. To reduce reection loss, anti-reection coating or surface texturing can be applied.

2.1.3 Scattering-matrix method

In real solar cell structures, there are usually several layers of dierent thicknesses, for example top single-layer or double-layer anti-reection coatings. In these cases, we can use the so-called scattering-matrix method to calculate recursively the total reectivity and to deduce the optical absorption in each layer. As illustrated in Figure 2.2, we consider N − 1 at layers of dierent materials (linear, homogeneous and isotropic). Each layer j (j = 1, 2, ..., N − 1) has a thickness hj and is described by their complex refractive indices nj. An incident plane wave comes from

the ambient medium (refractive index n0 = 1) with an angle θ0 to the surface normal. The

refracted angle in each layer j is denoted by θj. The substrate is assumed to be a semi-innite

medium of complex refractive index nN. The electric eld is denoted by Ej (resp. E0j) at top

(resp. bottom) of the layer j, and is represented by the superposition of two components: one propagating in the positive and one in the negative z direction, E+

j and E −

j , respectively.

The formalism of the scattering matrix method is to write the output eld as a function of the input eld, in the direction of propagation of the incident light (here top-down). The tangential component of the electric eld is continuous, leading to the Fresnel equations shown

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previously. From the linearity of the equations governing the propagation of the electric eld, matrix notations are introduced to conveniently express the relation that should be satised for the two components of the electric eld. Beginning with the most simple case with two semi-innite media separated by an interface (N = 1):

E0 0 + E₁− = S(1)E 0 0 − E₁+ =r01 t10 t01 r10 E0 0 − E₁+ (2.18)

where r01and t01 are the Fresnel complex reection and transmission coecients at the interface

between the medium 0 and the medium 1. These coecients depend on the complex refractive indices of the media, the angle of incidence and TE or TM polarization (see Equations2.15and

2.16). S(1) _{is the scattering matrix in the simple case of two-medium system (N = 1).}

Now we consider the case of j layers (j = 1, ..., N − 1) and assume that the problem to be solved up to the top of the layer j. Inside the layer j, the propagation of eld induces a phase change:

E0_j = E_j−exp(iδj)

E+_j = E_j0+exp(iδj)

(2.19) where δj = 2πnjhjcos(θj)/λ0 is related to the phase change and the attenuation (nj complex)

as the wave propagates through the layer j. Finally, the outgoing elds can be related to the incident elds by the scattering matrix S(N ) _{through a recursive algorithm [}₁₆_]:

E0 0 + E_N− = S(N )E 0 0 − E_N+ (2.20) Since no input eld is comping from the semi-innite substrate, we have E+

N = 0. Hence, we can

calculate explicitly the reection and transmission of light using the coecients of the matrix S(N ): Er Et =s11 s12 s21 s22 Ei 0 (2.21) As a consequence, the complex coecient for the reection of light is r = s11and the reectivity

is given by the square of the absolute value of r.

Considering the case of a single-layer anti-reection coating on a semiconductor wafer (N = 2), we can write the reectivity at normal incidence after manipulation of the scattering matrix S(2). R = r01+ r12exp(i2δ1) 1 + r01r12exp(i2δ1) 2 . (2.22)

We can thus suppress the reection of light by choosing r01 = r12 and exp(i2δ1) = −1. The

former leads to the choice of the refractive index n1 of the dielectric layer to be the square root

of that of the semiconductor: n1 =

√

n2 (n1 = 1.82.0 is relevant for general application), and

the latter results in the optimal thickness of the dielectric coating: h1=

λ0

4n1

. (2.23)

These conditions can be understood as a destruction interference of reected waves, and are strictly achieved at one wavelength. Further optimization of broadband anti-reection can be done using double-layer coating and graded optical indices.

(33)

Ein Hin kin 𝜃 nim nim nim nim substrate incident light n_i n_i n_i … … h1 hi hi+1 x z py dy p_x d_x x y x z _y (TE) n1 n_i+1 … … 1 2 0 -1 -2 -1 -2 0 1 2 Hin Ein kin 𝜃 x z _y (TM) (a) (b) (c) 3 -3

Figure 2.3 General description of a layer stack containing 2-dimensional periodic nanostructures used with Reticolo to calculate the electromagnetic eld inside the device. (a) Side-view (cross-section) of the device structure. Arrows with numbers 0, ±1, ±2... represent the orders of diraction. (b) Top-view of the nanostructured layer. The dashed frame delimits an unit cell of size px and py and the widths of the nanostructures are noted dx and dy. (c) Perspective views of a device with the axis orientation for an incident plane wave polarized in TE mode (left) and in TM mode (right). The incidence plane is shown in green.

2.1.4 RCWA for optical absorption

For optical absorption with more complex nanostructures like periodic diraction grating, we will employ numerical tools. In this thesis, we used the program Reticolo to perform 3-dimensional electromagnetic calculations for the diraction problem. It has been developed by Philippe Lalanne and Jean-Paul Hugonin, and Christophe Sauvan (IOGS, Institute d'Optique Graduate School) contributed to the customized software that was provided to our group. Reticolo is written in the Matlab language, and implements a frequency-domain modal method, known as the Rigorous Coupled-Wave Analysis (RCWA) [1719]. A free version of the software is available online.1

A general structure solved by Reticolo is shown in Figure 2.3. It is basically a multi-layer system (here: stacked vertically along the z-direction). One or several layers contains periodic 1-dimensional or 2-dimensional sub-wavelength patterns which have all identical periods in the x-and y-directions. Denition of 2-dimensional shapes other than rectangles (lozenges, polygons, circles, etc.) is also possible (parameter Ntre) using the convex envelop of several Cartesian patterns. We mainly focus on 2-dimensional gratings with the same period px and py and the

same nanostructure size dx and dy so that the solar cell is not polarization-dependent at normal

incidence.

In general, incoming light strikes on the top of the structure with an oblique incidence angle θ as shown in Figure2.3(c) for the incident plane wave polarized in the TE mode (left: electric eld perpendicular to the incidence plane) and for the incident plane wave polarized in the TM mode (right: magnetic eld perpendicular to the incidence plane). Calculation for the incident plane wave forming a non-zero azimuthal angle (conical diraction) is also possible, but the calculation time and computer memory increase considerably as the symmetry of the system decreases.

1_{Reticolo RCWA software (available online):} _{https://www.lp2n.institutoptique.fr/Membres-Services/}